Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

Marianne Akian 1, 2, 3 Stéphane Gaubert 1, 2, 3 Meisam Sharify 4
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, Polytechnique - X, CNRS - Centre National de la Recherche Scientifique : UMR
3 TROPICAL - TROPICAL
Inria Saclay - Ile de France
Abstract : We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and Pólya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated.
Type de document :
Article dans une revue
Linear Algebra and its Applications, Elsevier, 2017, 〈10.1016/j.laa.2016.11.004〉
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https://hal.inria.fr/hal-00881196
Contributeur : Marianne Akian <>
Soumis le : jeudi 7 novembre 2013 - 17:00:32
Dernière modification le : jeudi 11 janvier 2018 - 06:22:34

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Marianne Akian, Stéphane Gaubert, Meisam Sharify. Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots. Linear Algebra and its Applications, Elsevier, 2017, 〈10.1016/j.laa.2016.11.004〉. 〈hal-00881196〉

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